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ExampleConstruction: AI-generatedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26
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The support of the direct sum over all primes of Z/pZ is the set of all nonzero prime ideals of Z

Example

Let

M=⨁p primeZ/pZ.

Then

Supp⁡Z(M)={(p):p prime},

the set of all nonzero prime ideals of Z.

Facts & Assumptions

Given: The Z-module M=⨁p primeZ/pZ.

[L1]

The support of an arbitrary direct sum is the union of the supports of the summands (Support of an arbitrary direct sum is the union of the supports).

[L2]

The support of Z/pZ is exactly {(p)} (The support of a cyclic quotient is its vanishing set).

Verification

technique · direct
1.1

By [L1], Supp⁡Z(M)=⋃pSupp⁡Z(Z/pZ).

L1
2.1

Each summand contributes exactly the singleton {(p)} by [L2], so the union in step 1.1 is the set of all nonzero prime ideals of Z.

step 1.1L2
3.1

Hence Supp⁡Z(M)={(p):p prime}.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources